Casting shadows for transparent textured material
The problem seems to be the lighting. Both the ambient occlusion and the environment lighting are ignoring the transparent part of the outer cube, leaving the inner part completely shadowed.
To work around this, you could:

Use lamps. Lamps seem to illuminate the inner cube through the transparent texture fine.

Use cycles. The transparent part of the texture is lets light from environment lighting and AO as well as lamps through correctly when rendered with a node setup like this:


Bake AO in cycles (this is currently only available in development builds, but it will be in 2.71) and then use it in BI.

Model actual geometry for the outer hat instead of using an alpha texture.